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Instability analysis using component modes
Computers & S~rucrures Vol. 43, No. 3, pp. 451-457, Printed in Great Britain.
INSTABILITY
0045.7949/92 55.00 + 0.00 Pergamon Press Ltd
1992
ANALYSIS
USING COMPONENT
MODES
H. P. HUTTELMAIER Department of Engineering, Colorado School of Mines, Golden, CO 80401-1887,U.S.A. (Received 20
March
1991)
Abstract-This paper discusses substructuring techniques for instability and free vibration analyses within finite element applications. To date most of the techniques presented in the literature are applied to vibration problems. In this study the component mode method is applied for free vibrations and instability to plate bending problems and comparisons are made with regard to analytical solutions. It is pointed out that a lower degree of accuracy is obtained for instability problems as compared to corresponding vibration problems. In a separate study, instability frame problems using component modes are compared to other instability solution methods. Also, it is emphasized that substructuring techniques can be considered as attractive algorithms when using parallel computing environments.
INTRODUCTION
Substructuring or the use of reduced bases is extensively applied as an efficient solution strategy for large finite element systems. Applications in engineering analysis may be in the area of static, dynamic, stability, heat transfer analysis and others. During the last two to three decades many methods were developed and were specifically directed toward dynamic substructuring procedures. The need for lightweight structures such as thin shells, special space trusses, and general large space structures, or other large span systems has become apparent. Thus, emphasis is increasingly shifted from strength failure criteria to overall structural instability criteria demanding large scale solution procedures for instability problems. Ideally, a full geometrically nonlinear analysis should be applied to account for instability effects. This is usually computationally too intensive and not justifiable for most of the design problems at hand, thus an elastic stability procedure with efficient substructuring techniques is an attractive alternative. Moreover, some of these methods are very efficient when used within parallel algorithms. For a static analysis, substructuring means simply reduction of the system through an efficiently directed Gaussian elimination procedure and within numerical conditioning the results are always exact with regard to the full model. In a more complex analysis as for example in a dynamic or instability analysis the solution is typically obtained via a reduced eigenvalue problem and is always approximate. In a stability analysis the second coefficient matrix in the eigenvalue problem is the geometric stiffness or stress matrix which for the beam element, for example, is derived from first derivatives of displacement shapes. The geometric stiffness matrix can be indefinite (and singular), whereas the mass matrix, used in vibration problems is semi-positive. Some special features are inherent to buckling analyses which are particularly 451
relevant to substructuring: (i) in most cases only the lowest buckling mode is needed, and (ii) the possibility of different buckling load configurations within the structural assembly, which differs from a dynamic analysis where inertia properties remain usually unchanged throughout the structure. Due to the second fact a stability analysis is in many cases preceded by a substructured statics pass. In addition, for complicated structural configurations many different load patterns have to be checked which makes an elastic buckling analysis the more attractive method as opposed to a nonlinear analysis. In stability substructuring mostly static-condensation-type reductions are applied [l-3]. A different approach applied to columns and frames is described in [4]; in this case static deflection shapes due to sinusoidal loading patterns are derived and used as base vectors to form the transformation matrix for the reduction process. This idea is similar to a one-step subspace iteration with the difference that an initial ‘buckling-type’ loading is applied instead of an inertia loading. A reduction procedure which extends into postbuckling instability is discussed in [5]. In this study some of the techniques which are presently applied to dynamic substructuring only are investigated for instability problems. During the first stage of this study the component mode synthesis is applied and some comparisons between a stability and free vibration analysis are made in terms of accuracy and numerical efficiency. Different reduction procedures and main features of the component mode synthesis are discussed below.
DISCUSSION OF REDUCTION PROCEDURES
The crucial test when implementing reduction techniques is how well the original eigenvalue problem WI -~[GllIol
= 101
(1)
approximated by the reduced eigenvalue problem of the substructured model
THE COMPONEW
MODE SYNTHESIS FOR STABILITY PROBLEMS
A component mode synthesis method provides the WI -
PMIW = PI.
(2) means to reduce the global system from eqn (I) into
In the case of a buckling analysis the coefficient matrices [KJ and [G] are representing the stiffness and geometric stiffness, p and {u} are the buckling load with corresponding mode shape, respectively. Barred quantities, e.g., [a refer to the reduced system. If stability and inertia effects are combined (dynamic instability) the reduced eigenvalue problem is expressed as:
rrKl-
&PI
-
~*m.llw = PIT
(3)
where PO, w, and [il;il are the initial load the natural frequency and the reduced mass matrix, respectively. Major reduction or substructuring methods [used to reduce eqn (1) to eqn (2)] were mainly developed for dynamic substructuring and are summarized as follows: (a) Static condensation or Guyan reduction [6], where the ‘static deflection shapes’ are imposed through a transformation onto the inertial properties of the structure. This is equivalent to the application of substructure shape functions expressed in a discrete fo~ulation in terms of nodal degrees of freedom. (b) Iterative static condensation. In this case it is recognized that the transformation described in (a) depends on the natural frequency and an exact solution can essentially be closely approximated through an iterative re-evaluation of the eigenvalue problem [7,8]. 63 Component mode synthesis. This method can be visualized as a static condensation reduction scheme where the precision of the solution is enhanced by including selectively the naturalmode-energies from the substructure. Methods differ, depending on the nature of the imposed component constraints [9--l 11, as discussed below. (4 The use of Ritz vectors [ 121is usually limited to dynamic systems where the load application is known in advance. (4 Lumping procedures as applied frequently to the mass matrix are not substructuring procedures since the system is not reduced; however, a lumped or diagonalized mass matrix results at times in very effective solution schemes making substructuring unnecessary, particularly since diagonaIization is lost in the reduced substructure. It is noted however, that there is no lumping available for geometric stiffness matrices and different solution strategies are therefore required for an instability analysis.
an approximated much smaller system given in eqn (2). The method has developed into a variety of procedures and is best understood by studying the deformation modes a component can undergo. These modes are described in the literature (e.g. [13]) and are summarized as follows: (i) constraint modes which are generated through static condensation, (ii) normal modes, the eigenvectors of the internal degrees of freedom, (iii) attachment modes, used when coupling of free-free components is required; they are generated through unit load applications and represent therefore columns of the flexibility matrix, (iv) inertia-relief-modes are similar to attachment modes, however, a elf-equilibrating load set is applied instead of one unit load [14, 151. The physical interpretation for these modes is quite clear in a dynamic analysis, however this is not the case in a buckling analysis. During the reduction process the above modes or defo~ation shapes are applied in a usual Ritz analysis to reduce the complete system. In this study, mode types (i) and (ii) are applied in order to describe a ‘constraint component’. The physical interpretation of these modes in a buckling analysis is apparent, in addition this formulation results in substructure or macro elements, easily implemented in general purpose codes as well as in parallel procedures. A summary of the constraint component theory follows. Considering one component, subscript B refers to retained boundary nodes, subscript I stands for eliminated interior nodes and subscript N represents a transfo~ation by normal modes. The unreduced stiffness and geometric stiffness matrices are partitioned accordingly
Pa, bf A congruence transformation
is then performed
El = IW’WI~TIand @I= V’lT~lIll with the transfo~ation
@a, b)
matrix
(6) where [I] is the unit matrix. It is noted that the submatrices
Ua, W
453
Instability analysis using component modes represent a static condensation in the first matrix and store the m lowest buckling shapes {uli} of the constraint component in the second matrix. The buckling shapes are obtained from the internal eigenvalue problem
[&I - P,V%IH~,I= 101,
(8)
which is the local eigenvalue problem to be solved for each component and may be of small order, depending on the component size. Applying orthononnalized eigenvectors the transformation, results in the reduced component matrices
rGJB1 PII
Kl=
... ... Ph
and[’ = [
I
1
[&I
Pm1
[G,,]
[I]
.
NUMERICAL
Pa, W
The additional interior degrees of freedom represented by the m lowest eigenvalues (p,, to p,,,,) placed along the diagonal allow to approximate the internal ‘buckling energies’. A component or a substructure can therefore be of any dimension as long as all significant normal modes are included in the analysis. Matrix [GBN]= [GNBITderived through the transformation equation (5b) represents mixed modes. The reduced matrices [Kj and [Q can be obtained through an efficient stepwise algorith, [16, 171 which is represented in matrix form as follows: 14” =
Kl” - vL”[~~In~~xl
where {k}” and {g}, are the last column of the stiffness and geometric stiffness matrices, respectively, during the nth reduction step (see [17]), k,, and g,, are the n th diagonal elements. Equations (10) and (11) demonstrate that this reduction is always stable since only diagonal elements of the positive definite stiffness matrix appear in the denominator. Equations (10) and (11) can not be implemented to include normal mode degrees of freedom. The component matrices as shown in eqn (9) are assembled in the usual way into the global system which is then solved for global eigenvalues, hence the complete solution requires a solution of the eigenvalue problem on the local level for each component and once on the global level; for each case a subspace iteration is used in the present study.
(10)
Plate bending-free
EXAMPLES
vibration and instability analysis
In a first example a simply supported square plate is analyzed with the following geometry, and properties: 2 x 2 in, thickness = 0.1 in, modulus of elasticity = 3 x IO’ ksi, density = 7.33 x 10e5 kipssec2/in4, and Poisson’s ratio = 0.3. As for the instability analysis a unidirectional stress state of magnitude one is assumed. The exact solutions as indicated in Tables 1-3, are based on analytical results, obtained from Leissa [19] for free vibrations and from Timoshenko and Gere [20] for instability. A convergence study is performed using a 8 x 8 mesh (see Fig. 1) with incompatible plate bending elements. Three cases are studied: (i) the plate is used as one component, (ii) the plate is subdivided into
Table 1. Eigenvalues for simply supported square plate 8 x 8 mesh-one component Mode no. 2
1 % Error Stabilityanalysis Exact Full model
27.11 26.7
No. of normal modes 0 1 2 3 6
35.9 27.9 27.8 26.9 26.9
Vibrationanalysis Exact Full model
4808 4769
No. of normal modes 0 1 2 4 6
5647 4771 4771 4769 4769
1.5 34 4.3 4.1 <1
<1 18 <1
3 % Error
42.36 41.3 65.3 65.3 42.4 42.4 41.6 12020 11870 15560 15560 11890 11890 11880
2.5 58 58
1.2 31 31 <1
% Error 75.30 73.4
2.5
169.4 95.3 95.3 75.4 75.4
130 30 30 30 <1
12020 11870
1.2
15560 15560 15560 11890 11890
31 31 31
H. P. HUTTELMAIER
454
Table 2. Eigenvalues for simply supported square plate 8 x 8 mesh-two
qual components
Mode no.
1
2 % Error
Stabilityanalysis Exact Full model
27.11 26.7
1.5
3 % Error
% Error
42.36 41.3
2.5
75.30 73.4
2.5
53.18 41.83 41.83 41.83 41.83
29 1.2 1.2 1.2 1.2
108.30 77.62 77.62 74.40 74.10
47 5.7 5.7 1.3
12020 11870
1.2
12020 11870
1.2
No. of normal modes
0 131 2,2 333 494 Vibrationanalysis Exact Full model
27.25 26.73 26.73 26.72 26.72
1.9
4808 4769
<1
No. of normal modes
0 1,l 2,2 3.3
4832 4771 4771 4771
1.3
two even components, and (iii) the plate is subdivided into two uneven components. The results for the three cases are shown in Tables l-3, for an instability and vibration analysis. This demonstrates how many modes have to be included in order to obtain acceptable results. Comparisons are made with regard to analytical results (Exact) and the unreduced finite element model (Full model). It is noted that in particular for a single component structure at least one or in some cases several normal modes are needed for sufficient accuracy. For the two-component model a faster convergence is achieved since internal boundary nodes were retained. It is also noted that a vibration analysis converges faster and in some cases fewer normal modes are needed as compared to an instability analysis. A similar convergence behavior was exhibited by different mesh sizes. Table 3. Eigenvalues
12490 11880 11880 11870
5.2
13630 12900 11900 11900
15 8.6
The same plate example for a one component analysis is used in a dynamic stability study where the axial stress is varied from zero to a,, . These results are summarized in Fig. 2 where the normalized natural frequency is plotted versus the normalized axial stress; normalization is with regard to the full, unreduced finite element model. It is interesting to note that the convergence pattern in terms of natural mode participations is represented by a smooth transition from e = 0 to u,, . In this case, four normal modes are needed in order to achieve an accuracy of less than one per cent for the full range. Frame buckling
In a second study frame instability is investigated, first using a single story frame representing one component with the loading arrangement shown.
for simply supported square plate 8 x 8 mesh-two components
unequal
Mode no. 1
2 % Error
Stabilityanalysis Exact Full model No. of normal modes
27.11 26.7
0 1,l 292 3,3
27.14 26.84 26.80 26.78
Vibrarionanalysis Exact Full model
4808 4769
1.5
1.7 cl
il
3 % Error
% Error 42.36 41.3
2.5
75.30 73.4
2.5
47.28 42.10 42.05 41.87
14 1.9 1.8 1.3
99.14 79.34 74.89 74.43
35 8.1 2.0 1.4
12020 11870
1.2
12020 11870
1.2
No. of normal modes
0 131 232 3.3
4818 4775 4773 4770
1.0
12190 11890 11890 11880
2.7
13i50 12200 11920 11890
11 2.8 <1
Instability analysis using component modes
455
(4
Fig. 2. Dynamic instability curves for plate bending problem. (NM = internal normal modes; u/u,, = normalized axial stress, w/w, = normalized natural frequency.)
(‘4
(4 Fig. 1. Plate bending model for free vibration and instability. (a) One component. (b) Two equal components. (c) Two unequal components.
Support conditions are fixed-fixed (Fig. 3a) and fixed-hinged (Fig. 3b). The following properties are assumed: L = 10 units, stiffness EI = 1, and E/A = 1, where E, Z, and A are the modulus of elasticity, the moment of inertia, and the crosssectional area, respectively. Beam elements of equal stiffness and equal length for horizontal and vertical members are used, with ten elements per member. The exact solutions are taken from Bleich [21]. Table 4 shows convergence according to internal natural mode participation and using the two joint nodes as
boundary nodes. It is noted that the first frame eigenvalue is closely approximated when the eigenvalue problem is reduced to the number of degrees of freedom represented by the two joint nodes. To solve for larger eigenvalues, up to four internal modes are needed in order to achieve similar accuracy. In Fig. 4 the four internal natural modes are shown for the two frames. It is noted that they represent the basic buckling shapes of the individual members; two sets of multiple eigenvalues exist for the symmetric (fixed-fixed) frame. A three story frame assembled from three onestory components is shown in Fig. 5. In this convergence study it is assumed each component contributes the same number of internal natural modes. The results (see Table 5) indicate a faster convergence rate as compared to the single story frame, which is due to the larger number of reduced degrees of freedom. CONCLUSIONS
Out of several available substructuring schemes for dynamic analysis, the component mode synthesis is
P
P
P
P
4
4
4
A
I. : L
L
(a)
L
t
(b)
Fig. 3. Single story frame for instability problem. (a) Fixed-fixed. (b) Fixed-hinged.
0.07380 0.0743 1
0.04so3 0.04503
:
0.04698 0.04756
: 4
0.04779 0.04803 0.04770
No. of normal modes 0 0.04813 1 0.04808
Full model
Stability analysis Exact
0.04So7 0‘04S34
:
1.2
1.3
% Error
0.1581 0.1577
0.1584 0.1765
0.2080
O.lS60
0.4776 0.3226 0.2749 0.2719 0.2691
0.2517 0.2666
2
1.4
131.5
33
<1
79 21 3.1 2.0
5.5
% Error
Mode no.
1
< 1.0 1 < 1
1.2 1.1
< 1
% Error
0.07027 0.07031 0.07022
0.07068 0.0703 I
0.06740 0.06963
2
< 11.o <1
1.5 1.O
3.2
% Error
Mode no.
Table 5. Buckling toads for three story frame
0.04536
0.04482
0
No. of normal modes
Exact Full model
Fixed-hinged
No. of normal modes 0 0.07528 : 0.0749s 0.0749s 3 0.07482 4 0.07468
Full model
Fixed-fixed Exact
I
Table 4. Buckling loads for one story frame
0.09987 0.1003 0.9946
0.100s 0.1003
0.9865
0.2926 0.2931
0.2992 0.3982
0.5595
3
1.2 1.7
1.9 1.7
% Error
< 1
373.1
93
1.7
0.3194 0.3160 0.2902
120 52 2.7
% Error
0.7059 0.4776 0.3226
0.3141
3
00
0.417
fb)
0.215
(1)
B
Fig. 4. Internal buckling modes and corresponding eigenvalues for one story frame. (a) Fixed-fixed. (b) Fixed-hinged.
~
(2)
0.417
01 (1)
Instability analysis using component modes
451
4. B. S. Dong and J. A. Wolf, Stability analysis of
structures by a reduced system of generalized coordinates. Int. J. Solids Struct. 6, 1377-1388 (1970). 5. A. K. Noor and J. M. Peters, Recent advances in 6. 7. 8.
9.
Fig. 5. Three story frame for instability problem
10.
assessed with regard to instability analyses and its usefulness as well as the physical significance for some of the component buckling modes is demonstrated. A convergence study for plate vibration and plate instability is performed; this demonstrates that in general a higher accuracy is achieved for vibration problems and in some cases more internal natural modes are needed for an instability analysis. Domain decomposition methods such as the component mode synthesis are particularly attractive in parallel processing environments where the local (small sized) component eigenvalue problems are solved concurrently. This investigation is presently extended into a study using a shared memory parallel machine and the effectiveness of the method will be presented in a forthcoming paper.
17.
REFERENCES
18.
1. R. G. Anderson, B. M. Irons and 0. C. Zienkiewicz, Vibration and stability of plates using finite elements. ht. J. Solids Struct. 4, 1031-1055 (1968). 2. R. H. Gallagher, Finite Element Analysis Fundamentals. Prentice-Hall, Englewood Cliffs, NJ (1975). 3. W. Weaver and P. R. Johnston, Finite Elements for Structural Analysis. Prentice-Hall, Englewood Cliffs, NJ (1984).
11.
12.
13.
14. 15.
16.
19. 20. 21.
reduction methods for instability analysis of structures. Cornput. Struct. 16, 67-80 (1983). R. J. Guyan, Reduction of stiffness and mass matrices. AIAA JI 3, 380 (1965). M. Paz, Dynamic condensation method. AIAA JI 22, 724727 (1984). Y. T. Leung, An accurate method of dynamic substructuring with simplified computation. Int. J. Numer. Meth. Engng 14, 1241-1256 (1979). W. C. Hurty, Dynamic analysis of structural systems by component mode synthesis. AIAA JI 3, 678-685 (1965). W. C. Hurty, Introduction to modal synthesis techniques. Synthesis of vibrating systems. The Winter Annual Meeting of ASME, pp. I-13. Washington, D.C., Nov. (1976). W. A. Benfield and Hruda F. R. , Vibration analysis of structures by component substitution. AIAA JI 9, 125551261 (1971). E. L. Wilson, M. W. Yuan and J. M. Dickens, Dynamic analysis by direct superposition of Ritz vectors, Earthquake Engng Struct. Dyn. 10, 813-821 (1982). R. R. Craig and Chang C. J., On the use of attachment modes in substructure coupling for dynamic analysis. AIAA Paper 77405. Presented at AIAA/ASME/ ASCEIAHS 18th Structures Structural Dynamics, and Materials Conference (1977). R. H. MacNeal, A hybrid method of component mode synthesis. Cornput. Struct. 1, 581-601 (1971). S. Rubin, Improved component-mode representation for structural dynamic analysis. AIAA JI 13, 995-1006 (1975). E. L. Wilson, The static condensation algorithm. Int. J. Numer. Meth. Engng 8, 199-203 (1974). 0. A. Pekau and H. P. Huttelmaier, A versatile panel element for the analysis of shear wall structures. Comput. Struct. 12, 349-359 (1980). K. J. Bathe, Solution methods for large generalized eigenvalue problems. Report UC SESM 71-20, Civil Engineering Department, University of California, Berkeley, CA (1971). A. W. Leissa, Vibration of plates. NASA Report, SP-160, Washington, D.C. (1969). S. P. Timoshenko and J. M. Gere, Theory of Elastic Stability. McGraw-Hill (1961). F. Bleich, Buckling Strength of Metal Structures. McGraw-Hill (1961).
INSTABILITY
0045.7949/92 55.00 + 0.00 Pergamon Press Ltd
1992
ANALYSIS
USING COMPONENT
MODES
H. P. HUTTELMAIER Department of Engineering, Colorado School of Mines, Golden, CO 80401-1887,U.S.A. (Received 20
March
1991)
Abstract-This paper discusses substructuring techniques for instability and free vibration analyses within finite element applications. To date most of the techniques presented in the literature are applied to vibration problems. In this study the component mode method is applied for free vibrations and instability to plate bending problems and comparisons are made with regard to analytical solutions. It is pointed out that a lower degree of accuracy is obtained for instability problems as compared to corresponding vibration problems. In a separate study, instability frame problems using component modes are compared to other instability solution methods. Also, it is emphasized that substructuring techniques can be considered as attractive algorithms when using parallel computing environments.
INTRODUCTION
Substructuring or the use of reduced bases is extensively applied as an efficient solution strategy for large finite element systems. Applications in engineering analysis may be in the area of static, dynamic, stability, heat transfer analysis and others. During the last two to three decades many methods were developed and were specifically directed toward dynamic substructuring procedures. The need for lightweight structures such as thin shells, special space trusses, and general large space structures, or other large span systems has become apparent. Thus, emphasis is increasingly shifted from strength failure criteria to overall structural instability criteria demanding large scale solution procedures for instability problems. Ideally, a full geometrically nonlinear analysis should be applied to account for instability effects. This is usually computationally too intensive and not justifiable for most of the design problems at hand, thus an elastic stability procedure with efficient substructuring techniques is an attractive alternative. Moreover, some of these methods are very efficient when used within parallel algorithms. For a static analysis, substructuring means simply reduction of the system through an efficiently directed Gaussian elimination procedure and within numerical conditioning the results are always exact with regard to the full model. In a more complex analysis as for example in a dynamic or instability analysis the solution is typically obtained via a reduced eigenvalue problem and is always approximate. In a stability analysis the second coefficient matrix in the eigenvalue problem is the geometric stiffness or stress matrix which for the beam element, for example, is derived from first derivatives of displacement shapes. The geometric stiffness matrix can be indefinite (and singular), whereas the mass matrix, used in vibration problems is semi-positive. Some special features are inherent to buckling analyses which are particularly 451
relevant to substructuring: (i) in most cases only the lowest buckling mode is needed, and (ii) the possibility of different buckling load configurations within the structural assembly, which differs from a dynamic analysis where inertia properties remain usually unchanged throughout the structure. Due to the second fact a stability analysis is in many cases preceded by a substructured statics pass. In addition, for complicated structural configurations many different load patterns have to be checked which makes an elastic buckling analysis the more attractive method as opposed to a nonlinear analysis. In stability substructuring mostly static-condensation-type reductions are applied [l-3]. A different approach applied to columns and frames is described in [4]; in this case static deflection shapes due to sinusoidal loading patterns are derived and used as base vectors to form the transformation matrix for the reduction process. This idea is similar to a one-step subspace iteration with the difference that an initial ‘buckling-type’ loading is applied instead of an inertia loading. A reduction procedure which extends into postbuckling instability is discussed in [5]. In this study some of the techniques which are presently applied to dynamic substructuring only are investigated for instability problems. During the first stage of this study the component mode synthesis is applied and some comparisons between a stability and free vibration analysis are made in terms of accuracy and numerical efficiency. Different reduction procedures and main features of the component mode synthesis are discussed below.
DISCUSSION OF REDUCTION PROCEDURES
The crucial test when implementing reduction techniques is how well the original eigenvalue problem WI -~[GllIol
= 101
(1)
approximated by the reduced eigenvalue problem of the substructured model
THE COMPONEW
MODE SYNTHESIS FOR STABILITY PROBLEMS
A component mode synthesis method provides the WI -
PMIW = PI.
(2) means to reduce the global system from eqn (I) into
In the case of a buckling analysis the coefficient matrices [KJ and [G] are representing the stiffness and geometric stiffness, p and {u} are the buckling load with corresponding mode shape, respectively. Barred quantities, e.g., [a refer to the reduced system. If stability and inertia effects are combined (dynamic instability) the reduced eigenvalue problem is expressed as:
rrKl-
&PI
-
~*m.llw = PIT
(3)
where PO, w, and [il;il are the initial load the natural frequency and the reduced mass matrix, respectively. Major reduction or substructuring methods [used to reduce eqn (1) to eqn (2)] were mainly developed for dynamic substructuring and are summarized as follows: (a) Static condensation or Guyan reduction [6], where the ‘static deflection shapes’ are imposed through a transformation onto the inertial properties of the structure. This is equivalent to the application of substructure shape functions expressed in a discrete fo~ulation in terms of nodal degrees of freedom. (b) Iterative static condensation. In this case it is recognized that the transformation described in (a) depends on the natural frequency and an exact solution can essentially be closely approximated through an iterative re-evaluation of the eigenvalue problem [7,8]. 63 Component mode synthesis. This method can be visualized as a static condensation reduction scheme where the precision of the solution is enhanced by including selectively the naturalmode-energies from the substructure. Methods differ, depending on the nature of the imposed component constraints [9--l 11, as discussed below. (4 The use of Ritz vectors [ 121is usually limited to dynamic systems where the load application is known in advance. (4 Lumping procedures as applied frequently to the mass matrix are not substructuring procedures since the system is not reduced; however, a lumped or diagonalized mass matrix results at times in very effective solution schemes making substructuring unnecessary, particularly since diagonaIization is lost in the reduced substructure. It is noted however, that there is no lumping available for geometric stiffness matrices and different solution strategies are therefore required for an instability analysis.
an approximated much smaller system given in eqn (2). The method has developed into a variety of procedures and is best understood by studying the deformation modes a component can undergo. These modes are described in the literature (e.g. [13]) and are summarized as follows: (i) constraint modes which are generated through static condensation, (ii) normal modes, the eigenvectors of the internal degrees of freedom, (iii) attachment modes, used when coupling of free-free components is required; they are generated through unit load applications and represent therefore columns of the flexibility matrix, (iv) inertia-relief-modes are similar to attachment modes, however, a elf-equilibrating load set is applied instead of one unit load [14, 151. The physical interpretation for these modes is quite clear in a dynamic analysis, however this is not the case in a buckling analysis. During the reduction process the above modes or defo~ation shapes are applied in a usual Ritz analysis to reduce the complete system. In this study, mode types (i) and (ii) are applied in order to describe a ‘constraint component’. The physical interpretation of these modes in a buckling analysis is apparent, in addition this formulation results in substructure or macro elements, easily implemented in general purpose codes as well as in parallel procedures. A summary of the constraint component theory follows. Considering one component, subscript B refers to retained boundary nodes, subscript I stands for eliminated interior nodes and subscript N represents a transfo~ation by normal modes. The unreduced stiffness and geometric stiffness matrices are partitioned accordingly
Pa, bf A congruence transformation
is then performed
El = IW’WI~TIand @I= V’lT~lIll with the transfo~ation
@a, b)
matrix
(6) where [I] is the unit matrix. It is noted that the submatrices
Ua, W
453
Instability analysis using component modes represent a static condensation in the first matrix and store the m lowest buckling shapes {uli} of the constraint component in the second matrix. The buckling shapes are obtained from the internal eigenvalue problem
[&I - P,V%IH~,I= 101,
(8)
which is the local eigenvalue problem to be solved for each component and may be of small order, depending on the component size. Applying orthononnalized eigenvectors the transformation, results in the reduced component matrices
rGJB1 PII
Kl=
... ... Ph
and[’ = [
I
1
[&I
Pm1
[G,,]
[I]
.
NUMERICAL
Pa, W
The additional interior degrees of freedom represented by the m lowest eigenvalues (p,, to p,,,,) placed along the diagonal allow to approximate the internal ‘buckling energies’. A component or a substructure can therefore be of any dimension as long as all significant normal modes are included in the analysis. Matrix [GBN]= [GNBITderived through the transformation equation (5b) represents mixed modes. The reduced matrices [Kj and [Q can be obtained through an efficient stepwise algorith, [16, 171 which is represented in matrix form as follows: 14” =
Kl” - vL”[~~In~~xl
where {k}” and {g}, are the last column of the stiffness and geometric stiffness matrices, respectively, during the nth reduction step (see [17]), k,, and g,, are the n th diagonal elements. Equations (10) and (11) demonstrate that this reduction is always stable since only diagonal elements of the positive definite stiffness matrix appear in the denominator. Equations (10) and (11) can not be implemented to include normal mode degrees of freedom. The component matrices as shown in eqn (9) are assembled in the usual way into the global system which is then solved for global eigenvalues, hence the complete solution requires a solution of the eigenvalue problem on the local level for each component and once on the global level; for each case a subspace iteration is used in the present study.
(10)
Plate bending-free
EXAMPLES
vibration and instability analysis
In a first example a simply supported square plate is analyzed with the following geometry, and properties: 2 x 2 in, thickness = 0.1 in, modulus of elasticity = 3 x IO’ ksi, density = 7.33 x 10e5 kipssec2/in4, and Poisson’s ratio = 0.3. As for the instability analysis a unidirectional stress state of magnitude one is assumed. The exact solutions as indicated in Tables 1-3, are based on analytical results, obtained from Leissa [19] for free vibrations and from Timoshenko and Gere [20] for instability. A convergence study is performed using a 8 x 8 mesh (see Fig. 1) with incompatible plate bending elements. Three cases are studied: (i) the plate is used as one component, (ii) the plate is subdivided into
Table 1. Eigenvalues for simply supported square plate 8 x 8 mesh-one component Mode no. 2
1 % Error Stabilityanalysis Exact Full model
27.11 26.7
No. of normal modes 0 1 2 3 6
35.9 27.9 27.8 26.9 26.9
Vibrationanalysis Exact Full model
4808 4769
No. of normal modes 0 1 2 4 6
5647 4771 4771 4769 4769
1.5 34 4.3 4.1 <1
<1 18 <1
3 % Error
42.36 41.3 65.3 65.3 42.4 42.4 41.6 12020 11870 15560 15560 11890 11890 11880
2.5 58 58
1.2 31 31 <1
% Error 75.30 73.4
2.5
169.4 95.3 95.3 75.4 75.4
130 30 30 30 <1
12020 11870
1.2
15560 15560 15560 11890 11890
31 31 31
H. P. HUTTELMAIER
454
Table 2. Eigenvalues for simply supported square plate 8 x 8 mesh-two
qual components
Mode no.
1
2 % Error
Stabilityanalysis Exact Full model
27.11 26.7
1.5
3 % Error
% Error
42.36 41.3
2.5
75.30 73.4
2.5
53.18 41.83 41.83 41.83 41.83
29 1.2 1.2 1.2 1.2
108.30 77.62 77.62 74.40 74.10
47 5.7 5.7 1.3
12020 11870
1.2
12020 11870
1.2
No. of normal modes
0 131 2,2 333 494 Vibrationanalysis Exact Full model
27.25 26.73 26.73 26.72 26.72
1.9
4808 4769
<1
No. of normal modes
0 1,l 2,2 3.3
4832 4771 4771 4771
1.3
two even components, and (iii) the plate is subdivided into two uneven components. The results for the three cases are shown in Tables l-3, for an instability and vibration analysis. This demonstrates how many modes have to be included in order to obtain acceptable results. Comparisons are made with regard to analytical results (Exact) and the unreduced finite element model (Full model). It is noted that in particular for a single component structure at least one or in some cases several normal modes are needed for sufficient accuracy. For the two-component model a faster convergence is achieved since internal boundary nodes were retained. It is also noted that a vibration analysis converges faster and in some cases fewer normal modes are needed as compared to an instability analysis. A similar convergence behavior was exhibited by different mesh sizes. Table 3. Eigenvalues
12490 11880 11880 11870
5.2
13630 12900 11900 11900
15 8.6
The same plate example for a one component analysis is used in a dynamic stability study where the axial stress is varied from zero to a,, . These results are summarized in Fig. 2 where the normalized natural frequency is plotted versus the normalized axial stress; normalization is with regard to the full, unreduced finite element model. It is interesting to note that the convergence pattern in terms of natural mode participations is represented by a smooth transition from e = 0 to u,, . In this case, four normal modes are needed in order to achieve an accuracy of less than one per cent for the full range. Frame buckling
In a second study frame instability is investigated, first using a single story frame representing one component with the loading arrangement shown.
for simply supported square plate 8 x 8 mesh-two components
unequal
Mode no. 1
2 % Error
Stabilityanalysis Exact Full model No. of normal modes
27.11 26.7
0 1,l 292 3,3
27.14 26.84 26.80 26.78
Vibrarionanalysis Exact Full model
4808 4769
1.5
1.7 cl
il
3 % Error
% Error 42.36 41.3
2.5
75.30 73.4
2.5
47.28 42.10 42.05 41.87
14 1.9 1.8 1.3
99.14 79.34 74.89 74.43
35 8.1 2.0 1.4
12020 11870
1.2
12020 11870
1.2
No. of normal modes
0 131 232 3.3
4818 4775 4773 4770
1.0
12190 11890 11890 11880
2.7
13i50 12200 11920 11890
11 2.8 <1
Instability analysis using component modes
455
(4
Fig. 2. Dynamic instability curves for plate bending problem. (NM = internal normal modes; u/u,, = normalized axial stress, w/w, = normalized natural frequency.)
(‘4
(4 Fig. 1. Plate bending model for free vibration and instability. (a) One component. (b) Two equal components. (c) Two unequal components.
Support conditions are fixed-fixed (Fig. 3a) and fixed-hinged (Fig. 3b). The following properties are assumed: L = 10 units, stiffness EI = 1, and E/A = 1, where E, Z, and A are the modulus of elasticity, the moment of inertia, and the crosssectional area, respectively. Beam elements of equal stiffness and equal length for horizontal and vertical members are used, with ten elements per member. The exact solutions are taken from Bleich [21]. Table 4 shows convergence according to internal natural mode participation and using the two joint nodes as
boundary nodes. It is noted that the first frame eigenvalue is closely approximated when the eigenvalue problem is reduced to the number of degrees of freedom represented by the two joint nodes. To solve for larger eigenvalues, up to four internal modes are needed in order to achieve similar accuracy. In Fig. 4 the four internal natural modes are shown for the two frames. It is noted that they represent the basic buckling shapes of the individual members; two sets of multiple eigenvalues exist for the symmetric (fixed-fixed) frame. A three story frame assembled from three onestory components is shown in Fig. 5. In this convergence study it is assumed each component contributes the same number of internal natural modes. The results (see Table 5) indicate a faster convergence rate as compared to the single story frame, which is due to the larger number of reduced degrees of freedom. CONCLUSIONS
Out of several available substructuring schemes for dynamic analysis, the component mode synthesis is
P
P
P
P
4
4
4
A
I. : L
L
(a)
L
t
(b)
Fig. 3. Single story frame for instability problem. (a) Fixed-fixed. (b) Fixed-hinged.
0.07380 0.0743 1
0.04so3 0.04503
:
0.04698 0.04756
: 4
0.04779 0.04803 0.04770
No. of normal modes 0 0.04813 1 0.04808
Full model
Stability analysis Exact
0.04So7 0‘04S34
:
1.2
1.3
% Error
0.1581 0.1577
0.1584 0.1765
0.2080
O.lS60
0.4776 0.3226 0.2749 0.2719 0.2691
0.2517 0.2666
2
1.4
131.5
33
<1
79 21 3.1 2.0
5.5
% Error
Mode no.
1
< 1.0 1 < 1
1.2 1.1
< 1
% Error
0.07027 0.07031 0.07022
0.07068 0.0703 I
0.06740 0.06963
2
< 11.o <1
1.5 1.O
3.2
% Error
Mode no.
Table 5. Buckling toads for three story frame
0.04536
0.04482
0
No. of normal modes
Exact Full model
Fixed-hinged
No. of normal modes 0 0.07528 : 0.0749s 0.0749s 3 0.07482 4 0.07468
Full model
Fixed-fixed Exact
I
Table 4. Buckling loads for one story frame
0.09987 0.1003 0.9946
0.100s 0.1003
0.9865
0.2926 0.2931
0.2992 0.3982
0.5595
3
1.2 1.7
1.9 1.7
% Error
< 1
373.1
93
1.7
0.3194 0.3160 0.2902
120 52 2.7
% Error
0.7059 0.4776 0.3226
0.3141
3
00
0.417
fb)
0.215
(1)
B
Fig. 4. Internal buckling modes and corresponding eigenvalues for one story frame. (a) Fixed-fixed. (b) Fixed-hinged.
~
(2)
0.417
01 (1)
Instability analysis using component modes
451
4. B. S. Dong and J. A. Wolf, Stability analysis of
structures by a reduced system of generalized coordinates. Int. J. Solids Struct. 6, 1377-1388 (1970). 5. A. K. Noor and J. M. Peters, Recent advances in 6. 7. 8.
9.
Fig. 5. Three story frame for instability problem
10.
assessed with regard to instability analyses and its usefulness as well as the physical significance for some of the component buckling modes is demonstrated. A convergence study for plate vibration and plate instability is performed; this demonstrates that in general a higher accuracy is achieved for vibration problems and in some cases more internal natural modes are needed for an instability analysis. Domain decomposition methods such as the component mode synthesis are particularly attractive in parallel processing environments where the local (small sized) component eigenvalue problems are solved concurrently. This investigation is presently extended into a study using a shared memory parallel machine and the effectiveness of the method will be presented in a forthcoming paper.
17.
REFERENCES
18.
1. R. G. Anderson, B. M. Irons and 0. C. Zienkiewicz, Vibration and stability of plates using finite elements. ht. J. Solids Struct. 4, 1031-1055 (1968). 2. R. H. Gallagher, Finite Element Analysis Fundamentals. Prentice-Hall, Englewood Cliffs, NJ (1975). 3. W. Weaver and P. R. Johnston, Finite Elements for Structural Analysis. Prentice-Hall, Englewood Cliffs, NJ (1984).
11.
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14. 15.
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19. 20. 21.
reduction methods for instability analysis of structures. Cornput. Struct. 16, 67-80 (1983). R. J. Guyan, Reduction of stiffness and mass matrices. AIAA JI 3, 380 (1965). M. Paz, Dynamic condensation method. AIAA JI 22, 724727 (1984). Y. T. Leung, An accurate method of dynamic substructuring with simplified computation. Int. J. Numer. Meth. Engng 14, 1241-1256 (1979). W. C. Hurty, Dynamic analysis of structural systems by component mode synthesis. AIAA JI 3, 678-685 (1965). W. C. Hurty, Introduction to modal synthesis techniques. Synthesis of vibrating systems. The Winter Annual Meeting of ASME, pp. I-13. Washington, D.C., Nov. (1976). W. A. Benfield and Hruda F. R. , Vibration analysis of structures by component substitution. AIAA JI 9, 125551261 (1971). E. L. Wilson, M. W. Yuan and J. M. Dickens, Dynamic analysis by direct superposition of Ritz vectors, Earthquake Engng Struct. Dyn. 10, 813-821 (1982). R. R. Craig and Chang C. J., On the use of attachment modes in substructure coupling for dynamic analysis. AIAA Paper 77405. Presented at AIAA/ASME/ ASCEIAHS 18th Structures Structural Dynamics, and Materials Conference (1977). R. H. MacNeal, A hybrid method of component mode synthesis. Cornput. Struct. 1, 581-601 (1971). S. Rubin, Improved component-mode representation for structural dynamic analysis. AIAA JI 13, 995-1006 (1975). E. L. Wilson, The static condensation algorithm. Int. J. Numer. Meth. Engng 8, 199-203 (1974). 0. A. Pekau and H. P. Huttelmaier, A versatile panel element for the analysis of shear wall structures. Comput. Struct. 12, 349-359 (1980). K. J. Bathe, Solution methods for large generalized eigenvalue problems. Report UC SESM 71-20, Civil Engineering Department, University of California, Berkeley, CA (1971). A. W. Leissa, Vibration of plates. NASA Report, SP-160, Washington, D.C. (1969). S. P. Timoshenko and J. M. Gere, Theory of Elastic Stability. McGraw-Hill (1961). F. Bleich, Buckling Strength of Metal Structures. McGraw-Hill (1961).